VOLUME 79, NUMBER 24
PHYSICAL REVIEW LETTERS
15 DECEMBER 1997
Comment on ‘‘Dynamic Control of Cardiac
Alternans’’
In a recent Letter, Hall et al. [1] demonstrated that a
feedback control method can be used to suppress alternans
rhythms in a piece of cardiac tissue. Here we point
out that the method employed is properly viewed as a
modification of a previously analyzed time-delay feedback
method and that comparison of the two immediately
yields a surprising and potentially significant result.
From the perspective of linear control theory, the
system studied by Hall et al. can be modeled as a generic
dynamical system with a return map having an unstable
fixed point X p . In the linear vicinity of X p the equation
governing the system with feedback applied as
(1)
Xn11 ­ 2mXn 1 dln ,
(2)
dln ­ Qn bsXn 2 Xn21 d ,
Ω
1 if bsXn 2 Xn21 d . 0
Qn ­
,
(3)
0 otherwise
where in terms of the parameters of Ref. [1] we have m ­
A and b ­ aBy2. (Equations (1)–(3) are equivalent to
Eq. (2) of Ref. [1] in the linear vicinity of X p , with Qn
restricting the application of control to positive dln as
discussed in Ref. [1]). We consider only the case m . 1,
since that is where the fixed point is unstable. When
Qn ­ 1 (or 0), we say the control is active (or inactive)
for one iteration. The problem is to understand which
values of b will render the fixed point at X p stable.
If the Qn factor is removed from Eq. (2) so that control
is always active, the method is identical to the discrete
time-delay feedback discussed in Ref. [2]. In this case, it
is known that the control is effective only for sm 2 1dy2 ,
b , 1. (See Ref. [2], however, for a discussion of how
this domain can be enlarged using a simple recursive
algorithm.) Reference [1] notes that the introduction of
the Qn factor doubles the lower limit on b, but does not
mention an important feature of the upper boundary.
Figure 1 indicates the full domain of stability of the
fixed point of Eq. (1). It can be shown that only two
types of sequence Qn are possible as the system converges
to the fixed point: periodic repetition of either 01 . . . or
001 . . . , where the dots indicate a string of k 1’s with k
any integer. The infinite strips in Fig. 1 indicate regions
where sequences with k ­ 1, 2, or 3 will be observed
(in the absence of noise). The equations of the dark
curves that bound the k ­ 1 domain are b ­ m 2 1 and
b ­ m 2 1 1 1ym. In general, the boundaries of the
domain of order k are solutions to polynomial equations
of order k, so an analytic solution is possible only up
to k ­ 3 [3]. It can be shown, however, that the limit
of the infinite sequence
of stable domains is just the line
p
b ­ m 1 2 1 2 m 1 1.
We note that the domain of control is actually enlarged
by the inclusion of the Qn factor, a surprising result given
4938
0031-9007y97y79(24)y4938(1)$10.00
FIG. 1. The domain of control for Eqs. (1) – (3). The fixed
point is stable within the regions labeled by the sequence of
control activations. The light lines indicate optimal convergence. The large dots indicate that stable regions of higher
order exist up to the limiting dashed line. The region outlined
by dotted lines shows where the system would be stable without the Qn factor [2].
that this factor models an inability to affect the system
on certain iterations. In particular, the domain extends
to arbitrarily large values of m, as compared to the case
where control is always on for which m . 3 cannot be
controlled with any choice of b. We also note that optimal
convergence rates are obtained by operating at any point
on a boundary between the two sequence types of order
k. At these points the system is directed to the fixed point
after a single period of the control activation sequence. For
k ­ 1, the boundary is given by b ­ m2 ysm 1 1d.
Finally, we emphasize that control schemes in which
feedback is based on comparison to previous states of the
system rather than on an absolute reference seem to provide an efficient method of tracking the fixed point while
system parameters slowly drift [1,2,4,5]. This makes a variety of methods based on time-delay feedback particularly
interesting for biological systems, where appropriate response to parameter drift is often highly desirable.
This work was supported by NSF and the Whitaker
Foundation.
Daniel J. Gauthier and Joshua E. S. Socolar
Physics Department and CNCS
Duke University
Durham, North Carolina 27708
Received 25 August 1997
[S0031-9007(97)04701-7]
PACS numbers: 87.22.– q, 05.45.+b, 07.05.Dz, 87.10.+e
[1] K. Hall et al., Phys. Rev. Lett. 78, 4518 (1997).
[2] J. Socolar, D. Sukow, and D. Gauthier, Phys. Rev. E 50,
3245 (1994).
[3] K. Hall (private communication) obtained these results
independently.
[4] D. Gauthier et al., Phys. Rev. E 50, 2343 (1994).
[5] S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. E
49, R971 (1994).
© 1997 The American Physical Society